\long\def\symbolfootnote[#1]#2{\begingroup%
\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup}

\newcommand{\Probab}[1]{\mbox{}{\mathbb{P}}\left[#1\right]}
\newcommand{\Exp}{\mbox{}{\mathbb{E}}}
\newcommand{\Expect}[1]{\mbox{}{\mathbb{E}}\left[#1\right]}
\newcommand{\Varnce}[1]{\mbox{}{\bf{Var}}\left[#1\right]}
\newcommand{\Trace }[1]{\mbox{}{\mathrm{Tr}}\left(#1\right)}
%\newcommand{\diag  }[1]{\mbox{}{\textrm{diag}}\left(#1\right)}
\newcommand{\Sqrt  }[1]{\mbox{}\left(#1\right)^{1/2}}
\newcommand{\Qdrt  }[1]{\mbox{}\left(#1\right)^{1/4}}
\newcommand{\FNormB }[1]{\mbox{}\left\|#1\right\|_{\mathrm{F}}  }
\newcommand{\FNormBS}[1]{\mbox{}\left\|#1\right\|_{\mathrm{F}}^2}
\newcommand{\FNorm }[1]{\mbox{}\|#1\|_\mathrm{F}  }
\newcommand{\FNormS}[1]{\mbox{}\|#1\|_\mathrm{F}^2}
\newcommand{\FNormF}[1]{\mbox{}\|#1\|_\mathrm{F}^4}
\newcommand{\FNormQ}[1]{\mbox{}\left\|#1\right\|_\mathrm{F}^4}
\newcommand{\TNormB }[1]{\mbox{}\left\|#1\right\|_2  }
\newcommand{\TNormBS}[1]{\mbox{}\left\|#1\right\|_2^2}
\newcommand{\TNorm }[1]{\mbox{}\|#1\|_2  }
\newcommand{\TNormS}[1]{\mbox{}\|#1\|_2^2}
\newcommand{\TNormF}[1]{\mbox{}\|#1\|_2^4}
\newcommand{\TNormQ}[1]{\mbox{}\left\|#1\right\|_2^4}
\newcommand{\XNormB }[1]{\mbox{}\left\|#1\right\|_{\xi}  }
\newcommand{\XNormBS}[1]{\mbox{}\left\|#1\right\|_{\xi}^2}
\newcommand{\XNorm }[1]{\mbox{}\|#1\|_{\xi}  }
\newcommand{\XNormS}[1]{\mbox{}\|#1\|_{\xi}^2}
\newcommand{\PNorm }[1]{\mbox{}\|#1\|_{p}  }
\newcommand{\PNormS}[1]{\mbox{}\|#1\|_{p}^2}
\newcommand{\XNormQ}[1]{\mbox{}\left\|#1\right\|_{\xi}^4}
\newcommand{\VTNorm }[1]{\mbox{}\left|#1\right|  }
\newcommand{\VTNormS}[1]{\mbox{}\left|#1\right|^2}
\newcommand{\VTNormQ}[1]{\mbox{}\left|#1\right|^4}
\newcommand{\VTTNorm }[1]{\mbox{}\left|#1\right|_2  }
\newcommand{\VTTNormS}[1]{\mbox{}\left|#1\right|_2^2}
\newcommand{\VTTNormQ}[1]{\mbox{}\left|#1\right|_2^4}
\newcommand{\MCNorm}[1]{\mbox{}\|#1\|_{1 \rightarrow 2}}
\newcommand{\MCNormS}[1]{\mbox{}\|#1\|_{1 \rightarrow 2}^2}
\newcommand{\INorm}[1]{\mbox{}\|#1\|_\infty}
\newcommand{\pinv}[1]{ {#1}^\dagger}

%\newtheorem{theorem}{\bf Theorem}[]
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{axiom}[theorem]{Axiom}
%\newtheorem{thm}[theorem]{\bf Theorem}

\newcommand{\qedsymb}{\hfill{\rule{2mm}{2mm}}}
\newcommand{\transp}{^{\textsc{T}}}
\newcommand{\trace}{\text{\rm Tr}}
\newcommand{\mat}[1]{{\ensuremath{\bm{\mathrm{#1}}}}}
\newcommand{\dvc}{\ensuremath{d_{\small \sf VC}}}
%\newcommand{\abs }[1]{\left|#1\right|}
\newcommand{\EE}[1]{\ensuremath{\mathbb{E}\left[#1\right] } }
\newcommand{\var}[1]{\text{Var}\ensuremath{\left[#1\right] } }
\newcommand{\lambdamax}[1]{\ensuremath{\lambda_{\mathrm{max}}\left(#1\right)}}
\renewcommand{\vec}[1]{\ensuremath{\bm{#1}}}

\def\gammab{{\bm{\gamma}}}
\def\kappab{{\bm{\kappa}}}
\def\sig{{\bm{\Sigma}}}
\def\sigplus{{\bm{\Sigma}^{+}}}
\def\siginv{{\bm{\Sigma}^{-1}}}
\def\bet{{\bm{\beta}}}
\def\one{{\bm{1}}}
\def\exp{\hbox{\rm exp}}
\def\rank{\hbox{\rm rank}}
\def\col{\hbox{\rm col}}
\def\ker{\hbox{\rm ker}}
\def\ahat{{\hat\a}}
\def\b{{\mathbf b}}
\def\e{{\mathbf e}}
\def\expe{{\mathrm e}}
\def\q{{\mathbf q}}
\def\rb{{\mathbf r}}
\def\s{{\mathbf s}}
\def\u{{\mathbf u}}
\def\v{{\mathbf v}}
\def\d{{\mathbf \delta}}
\def\xhat{{\hat\x}}
\def\yhat{{\hat\y}}
\def\A{\matA}
\def\B{\matB}
\def\C{\matC}
\def\Ahat{\hat\matA}
\def\Atilde{\tilde\matA}
\def\Btilde{\tilde\matB}
\def\Stilde{\tilde\matS}
\def\Utilde{\tilde\matU}
\def\Vtilde{\tilde\matV}
\def\E{{\cl E}}
\def\G{{\cl G}}
\def\hset{{\cl H}}
\def\Q{{\bm{Q}}}
\def\U{{\bm{U}}}
\def\V{{\bm{V}}}
\def\win{\hat{\w}}
\def\wopt{\w^*}
\def\matAhat{\hat\mat{A}}
\def\matA{\mat{A}}
\def\matB{\mat{B}}
\def\matC{\mat{C}}
\def\matD{\mat{D}}
\def\matE{\mat{E}}
\def\matG{\mat{G}}
\def\matH{\mat{H}}
\def\matI{\mat{I}}
\def\matJ{\mat{J}}
\def\matK{\mat{K}}
\def\matL{\mat{L}}
\def\matN{\mat{N}}

\def\matM{\mat{M}}
\def\matP{\mat{P}}
\def\matQ{\mat{Q}}
\def\matR{\mat{R}}
\def\matS{\mat{S}}
\def\matT{\mat{T}}
\def\matU{\mat{U}}
\def\matV{\mat{V}}
\def\matW{\mat{W}}
\def\matX{\mat{X}}
\def\matY{\mat{Y}}
\def\matZ{\mat{Z}}
\def\matOmega{\mat{\Omega}}
\def\matSig{\mat{\Sigma}}
\def\matTh{\mat{\Theta}}
\def\matGam{\mat{\Gamma}}

\def\matOmega{\mat{\Omega}}
\def\matSig{\mat{\Sigma}}
\def\matTh{\mat{\Theta}}
\def\matGamma{\mat{\Gamma}}
\def\matPhi{\mat{\Phi}}
\def\matPsi{\mat{\Psi}}
\def\matDelta{\mat{\Delta}}
\def\matXi{\mat{\Xi}}

\def\MatPsi{\mat{\Psi}}
\def\w{{\mathbf{w}}}
\def\ein{{\cl E_{\text{\rm in}}}}
\def\eout{{\cl E}}
\def\scl{{\textsc{l}}}
\def\scu{{\textsc{u}}}
\def\phiu{{\overline{\phi}}}
\def\psiu{{\overline{\psi}}}
\def\phil{{\underbar{\math{\phi}}}}
\DeclareMathSymbol{\Prob}{\mathbin}{AMSb}{"50}
%\DeclareMathSymbol{\Exp}{\mathbin}{AMSb}{"45}
\newcommand\remove[1]{}

\def\nnz{{ \rm nnz }}

\def\mvec#1#2{[#1_1,\ldots ,#1_#2]}
\def\msum#1#2#3{$$\sm_{#1=#2}^#3$$}
\def\mint#1#2{$$\int_#1^#2$$}
\def\math#1{$#1$}
\def\mathd#1{$$#1\eqa$$}
\def\mand#1{$$#1$$}
%\def\v#1{{\mathbf #1}}
\def\frac#1#2{{#1\over #2}}
\def\ital#1{{\it #1}}
\def\mld#1{\begin{equation}
#1
\end{equation}}
\def\eqar#1{\begin{eqnarray}
#1
\end{eqnarray}}
\def\eqan#1{\begin{eqnarray*}
#1
\end{eqnarray*}}

\DeclareMathSymbol{\R}{\mathbin}{AMSb}{"52}

\def\Nat{{\mathbf N\hspace*{-1.89ex}\rule{0.07ex}{1.5ex}\hspace*{1.89ex}\hspace*{-1.7ex}\rule{0.1ex}{1.5ex}\hspace*{1.7ex}\hspace*{-.6ex}\rule{0.1ex}{1.5ex}\hspace*{.6ex}}}
\def\qed{\hfill\rule{2mm}{2mm}}
\def\sm{\sum\limits}
\def\choose#1#2{\left({{#1}\atop{#2}}\right)}
\def\cl#1{{\cal #1}}
\def\prob#1{{P\left[{#1}\right]}}
\def\mathbox#1{{\fbox{\math{#1}}}}
\def\mandbox#1{{\fbox{\math{\displaystyle #1}}}}
\def\argmin{\mathop{\hbox{argmin}}\limits}
\def\argmax{\mathop{\hbox{argmax}}\limits}

\def\mbf#1{{\mbox{\boldmath{$#1$}}}}
\def\h{{\mathbf h}}
\def\m{{\mathbf m}}
\def\x{{\mathbf x}}
\def\r{{\mathbf r}}
\def\y{{\mathbf y}}
\def\z{{\mathbf z}}
\def\X{{\mathbf X}}
\def\Y{{\mathbf Y}}
\def\a{{\mathbf a}}
\def\b{{\mathbf b}}
\def\exp#1{{\left\langle#1\right\rangle}}
\def\E#1{{{\mathbf E}\left[#1\right]}}
\def\norm#1{{\left\|#1\right\|}}
\def\mynorm#1{{\|#1\|}}
\def\floor#1{{\left\lfloor\,#1\,\right\rfloor}}
\def\ceil#1{{\left\lceil\,#1\,\right\rceil}}
\def\r#1{{(\ref{#1})}}
\def\rot#1{{\rotatebox{90}{#1\ }}}
